Harmonic model description of the Franck-Condon density for a betaine dye molecule

Franck-Condon (FC) factors and the FC density associated with an electron transfer reaction are calculated for a betaine molecule, pyridinium-N-phenoxide betaine (4-(1-pyridinio)phenolate) in its S, excited state. FC factors and density functions for harmonic vibrational modes are computed first by modifying the three level-fixed binary tree algorithm (Ruhoff, P. T.; Ratner, M. A. Int. J. Quant. Chem. 2000, 1, 383), a sum-overstates method based on recursion relations. This modified method allows the calculation of FC factors for 60 vibrational modes and avoids memory problems due to the large number of modes. The effects on the FC density of frequency shifts and mode mixing (Duschinky rotation) are included. For comparison, the more efficient time-dependent alternative is also employed for the calculation of the FC density function for the harmonic motion. In all cases, for a torsional motion which cannot be described by a harmonic potential, the FC density function is computed through the time-dependent method. We show that the sum-over-states method agrees well with the time-dependent method except for the high-frequency region. There the sum-over-states method is inadequate even when greater than 10(14) FC factors are included. We find that both frequency shifts and Duschinsky rotation increase the number of FC factors in the high-frequency region, and as a result, they make the FC density function broader. It is shown that frequency shifts have the greater effect. In the high-frequency region we do not observe the strong exponential decay of the FC density function which characterizes the weak coupling limit (relatively small vibrational reorganization energy). We find that the betaine dye falls into the strong coupling limit. The fitting of the FC density function with a simple model which includes one classical degree of freedom and one high-frequency quantal degree of freedom and the comparison of the fitting parameters with comparable exact values show that the simple model provides reasonable physical values such as reorganization energies.